Question

what's the domain of 1/ 4th root (x^2-19x)?

Answer 1

\[f(x) = \frac{1}{\sqrt[4]{x^2-19x}}\]
This function has a domain of R, i.e. (-inf, +inf) except where the denominator becomes zero or negative, where the function is undefined.
The domain, excluding complex, is therefore
(-inf, 0) U (19,+inf)
since the denominator becomes <=0 on the interval [0,19].